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Mini-course: Polyhedral structures in algebraic geometry

MINI-COURSE: POLYHEDRAL STRUCTURES IN ALGEBRAIC GEOMETRY

Mini-course: Polyhedral structures in algebraic geometry

Abstract: Algebraic geometry studies the zero locus of polynomial equations connecting the related algebraic and geometrical structures. In several cases, nevertheless the theory is extremely precise and elegant, it is hard to read in a simple way the information behind such structures. A possible way of avoiding this problem is that of associating to polynomials some polyhedral structures that immediately give some of the information connected to the zero locus of the polynomial. In relation to this strategy I will introduce Newton-Okounkov bodies and Tropical Geometry, underlying the connection between the two theories.I will conclude stating a recent result in collaboration with E. Postinghel, where the interplay of tropical geometry and Newton-Okounkov bodies gives a flat degeneration for Mori Dream Spaces.

May 2018

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Attila Kovacs -

Foundational themes

FOUNDATIONAL THEMES

Foundational themes

Some themes of the last century research on foundations of logic and mathematics will be considered under the point of view of the current logical research: 1. Incompleteness theorems. 2. Constructive omega-rule. 3. Ordinal numbers and dilators

Generalized Symmetries from the Integrability of Difference Equations

Formal Symmetries and Integrable Lattice Equations

semester I

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Advanced Graphics for Scientific Data

ADVANCED GRAPHICS FOR SCIENTIFIC DATA

Advanced Graphics for Scientific Data

We will introduce some usefull tools for Scientific Data visualization, starting from GNUplot and MATLAB for drawing basic 1D and 2D functions, then moving to Paraview for exploring more complex features of 3D data sets

semester II

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K-THEORY IN CONDENSED MATTER PHYSICS

K-THEORY IN CONDENSED MATTER PHYSICS

K-THEORY IN CONDENSED MATTER PHYSICS

In the last decades, solid state physics has witnessed a plethora of phenomena that have a topological origin: from the pioneering works of Thouless and collaborators to explain the quantization of the transverse conductivity in the quantum Hall effect, to the thriving field of topological insulators and superconductors. This field of research rapidly attracted the attention of mathematical physicists as well.

The methods involved in the theoretical understanding of these phenomena, at least in the one-particle approximation, rely on the theory of vector bundles and K-theory. The course will therefore be divided in two parts:

the first part of the course will cover topics from differential topology and geometry, in particular the classification of vector bundles, their invariants, and K-theory;

the second part will be devoted to the physical applications, discussing the periodic table of topological insulators, and the relation of their topological labels with quantum transport.

Some basic differential geometry and a first course in the mathematics of quantum mechanics will be assumed as prerequisites.

Depending on the inclinations, background and interests of the students, and if time permits, more advanced topics could also be covered, including for example:

K-theory for C*-algebras and applications to disordered topological insulators;

obstruction theory and constructive algorithms for Wannier functions;

universality of the Hall conductivity with respect to weak interactions via renormalization group methods.